Percolation hyperscaling

Muthukumar [44] further investigated the effects of polydispersity, which are important for crosslinking systems. He used a hyperscaling relation from percolation theory to obtain his results. If the excluded volume is not screened, n is related to df by... 

As an illustration of the Rouse model, consider the polydisperse mixture of polymers produced by random branching with short chains between branch points. The molar mass distribution and size of the branched polymers in this critical percolation limit were discussed in Section 6.5. Close to the gel point, some very large branched polymers (with M> 10 ) are formed and the intuitive expectation is that such large branched polymers would be entangled. However, recall that hyperscaling requires... 

Use this equation for a general polydisperse fractal polymer and the dis-tribution function of Eq. (6.93) to rederive the Rouse stress relaxation modulus of the melt randomly branched polymers found in the three-dimensional percolation polymerization reactor [Eq. (8.147)]. Hint Use hyperscaling. 

For percolation, scaling and hyperscaling is used whenever it determines an exponent more accurately than direct data. Rational numbers indicate (presumably) exact results, numbers with a decimal point are numerical extrapolations with an estimated error typically of the order of one unit in the last digit given. Data from earlier reviews and recent research ... 

---